Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Q4CQ (page 402)

A scientifically untrained but curious friend asks, "When I walk into a room, is there a chance that all the air will be on the other side?" How do you answer this question?

Short Answer

Expert verified

One present only one microstate, mathematically it is one of infinity.

Step by step solution

01

microstates

Microstate is a term that describes the microscopic properties of a thermodynamic system

02

show, thatis there a chance that all the air will be on the other side?

Although it is theoretically possible, it is borderline non-realistic, so it is ought to be dismissed.

Of the infinitive microstates that air molecules can have in that room, this one present only one microstate, mathematically it is one of infinity.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

To obtain the Maxwell speed distribution, we assumed a uniform temperature. kinetic-only energy of E=mvx2+vy2+vz2, and we assumed that we wished to find the average of an arbitrary function of X. Along the way, we obtained probability per unit height speed,P(v).

a) Assuming a uniform temperature and an energy ofE=12mvx2+vy2+vz3+mgyand assuming we wish to find the average of an arbitrary function of Y, obtain a probability per unit height,P(y) .

b) Assuming a temperature of300K. how much less the density of the atmosphere'sat an altitude of(about3000ft) than at sea level'?

(c) What of theO2in the atmosphere?

We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy ! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system of11oscillators sharing a total energy of just5hω0 . In the symbols of Section 9.3. N=11andM=5 .

  1. Using equation(9-9) , calculate the probabilities ofn , being0,1,2, and3 .
  2. How many particlesNn , would be expected in each level? Round each to the nearest integer. (Happily. the number is still 11. and the energy still5hω0 .) What you have is a distribution of the energy that is as close to expectations is possible. given that numbers at each level in a real case are integers.
  3. Entropy is related to the number of microscopic ways the macro state can be obtained. and the number of ways of permuting particle labels withN0 ,N1,N2 , and N3fixed and totaling11 is11!(N0!N1!N2!N3!) . (See Appendix J for the proof.) Calculate the number of ways for your distribution.
  4. Calculate the number of ways if there were6 particles inn=0.5 inn=1 and none higher. Note that this also has the same total energy.
  5. Find at least one other distribution in which the 11 oscillators share the same energy, and calculate the number of ways.

  6. What do your finding suggests?

Discusses the energy balance in a white dwarf. The tendency to contract due to gravitational attraction is balanced by a kind of incompressibility of the electrons due to the exclusion principle.

(a) Matter contains protons and neutrons, which are also fanions. Why do the electrons become a hindrance to compression before the protons and neutrons do?

(b) Stars several times our Sun's mass has sufficient gravitational potential energy to collapse further than a white dwarf; they can force essentially all their matter to become neutrons (formed when electrons and protons combine). When they cool off, an energy balance is reached like that in the white dwarf but with the neutrons filling the role of the incompressible fermions. The result is a neutron star. Repeat the process of Exercise 89. but assume a body consisting solely of neutrons. Show that the equilibrium radius is given by

R=3ℏ22G(3π22mn8M)1/3

(c) Show that the radius of a neutron star whose mass is twice that of our Sun is only about10km .

There are more permutations of particle labels when two particles have energy0 and two have energy1 than when three particles have energy 0and one has energy . 2(The total energiesarethe same.) From this observation alone argue that the Boltzmann distribution should be lower than the Bose-Einstein at the lower energy level.

Consider a room divided by imaginary lines into three equal parts. Sketch a two-axis plot of the number of ways of arranging particles versus NleftandNrightfor the caseN=1023, Note that Nmiddleis not independent, being of courseN−Nnght−NleftYour axes should berole="math" localid="1658331658925" NleftandNright, and the number of ways should be represented by density of shading. (A form for numbers of ways applicable to a three-sided room is given in Appendix I. but the question can be answered without it.)
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Famous Physicists

Read Explanation

Torque and Rotational Motion

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Turning Points in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free